For \(k\in \mathbb {N}\) , let \((\chi _\nu )_{\nu }\)  be a partition of unity of \(\mathbb {R}^{d}\setminus \lbrace 0\rbrace \)  with smooth zero-homogeneous functions, which localize to cones of aperture aproximately \(2^{-k/2}\) , and let \(g\in \mathcal {S}(\mathbb {R})\)  be such that \(|g(t)|\ge 1\)  for \(t\in [0,1]\) , and \(\mathrm {supp}(\widehat{g}\,)\subseteq [-1,1]\) . After rescaling to unit frequencies (see e.g. [1]}), using that \(\Vert e^{it\sqrt{-\Delta }}f\Vert _{L^{p}([0,1]\times \mathbb {R}^{d})}\le \Vert g(t)e^{it\sqrt{-\Delta }}f\Vert _{L^{p}(\mathbb {R}\times \mathbb {R}^{d})}\)  and that \((t,x)\mapsto g(t)e^{it\sqrt{-\Delta }}f(x)\)  has frequency support near the light cone, it then follows from the \(\ell ^{2}\) -decoupling inequality [2]} that
\(\Vert e^{it \sqrt{- \Delta }}f \Vert _{L^p([0,1]\times \mathbb {R}^{d})} \lesssim 2^{k(\overline{s}(p) + \varepsilon )} \Big (\sum _\nu \Vert g(t)e^{it \sqrt{- \Delta }} \chi _\nu (D) f \Vert ^2_{L^p(\mathbb {R}\times \mathbb {R}^{d})} \Big )^{1/2}\) 
